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25Methods of computational algebra and solving mathematical physics equations 41
terms are made proportional to the weak local residuals originally introduced in [2]. The performance of the proposed schemes will be demonstrated on a number of challenging benchmarks.
References
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Jiang Y., Shu C..W., Zhang, M. An alternative formulation of finite difference weighted ENO schemes with Lax.Wendroff time discretization for conservation laws // SIAM J. Sci. Comput.. 2013.V. 35 (2).P. A1137.A1160.
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Karni S., Kurganov A. Local error analysis for approximate solutions of hyperbolic conservation laws // Adv. Comput. Math.V. 22.P. 79.99.
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Kurganov A., Liu Y. New adaptive artificial viscosity method for hyperbolic systems of conservation laws // J. Comput. Phys.2012. V. 231. P. 8114.8132.
Dualapproachasempiricalreliabilityforfractionaldifferentialequations
P. B. Dubovski, J. Slepoi
Stevens Institute of Technology Email: pdubovsk@stevens.edu
DOI 10.24412/cl.35065.2021.1.00.12
We consider fractional differential equations with derivatives in the Gerasimov.Caputo sense. Computa�tional methods for these equations exhibit essential instability. Even a minor modification of the coefficients or other entry data may switch good results to divergent. The goal of this talk is to suggest a reliable dual ap�proach which fixes this inconsistency. We suggest to use of two parallel computational methods based on the transformation of fractional derivatives through (1)integration by parts and (2) substitution method. The by.parts method is known whereas the substitution method is novel. We prove the stability theorem for the sub�stitution method and introduce a proper discretization scheme that fits the grid points for both methods. The solution is treated as reliable (robust) only if both schemes produce the same results. In order to demonstrate the proposed dual approach, we apply it to linear, quasilinear and semilinear equations and obtain very good precision. The provided examples and counterexamples support the necessity to use the dual approach be�cause either method, used separately, may produce incorrect results. The order of the exactness is close to the exactness of the approximations of fractional derivatives.
References
1.
Gerasimov A. N. A generalization of the deformation laws and its application to the problems of internal friction // Appl. Math.and Mechanics. 1948.V. 12.P. 251.260.
2.
Zhukovskaya N. V., Kilbas A. A. Solving homogeneous fractional differential equations of Euler type // Differential Equations. 2011.V. 47, N. 12.P. 1714.1725.
3.
Albadarneh R., M. Zerqat M., Batiha I. Numerical Solutions for linear and non.linear fractional differential equations // International J. of Pure and Applied Mathematics. 2016. V. 106, N. 3. P. 859.871.
4.
Kilbas A. A.,Srivastava H. M., Trujillo J. J. Theory and applications of Fractional Differential equations. Amsterdam: Elsevier Science 2006.
TheCFD.DEMnumericalmodelforsimulationofafluidflowwithlargeparticles
D. V. Esipov
Kutateladze Institute of Thermophysics SB RAS Email: denis@esipov.org
DOI 10.24412/cl.35065.2021.1.00.13
The mathematical modelfor considering flows consists of the Navier�Stokes equations for a viscous in�compressible fluid flow and a set of motion and rotation equations for every particle.The no.slip boundary condition posed at the surface of the particles connects these two different sets of equations. In the numerical
